Human structure-making flourishes in times of freedom and general well-being. One such time began in the 11th century and it lasted 200 years. The weather had begun warming. New power sources had been freeing hands and lighting up a darker age: first the properly harnessed horse, then widespread use of water wheels, finally the European windmill. A new civilization was taking root.

Buildings of the French abbey at Cluny began rising around 1050 AD. Pilgrimages to the Holy Land had brought back Eastern ideas about art, so Cluny leavened the stern architecture of Rome with oriental exotica.

And here, two people enter our story: The aristocratic St. Bernard left the world to become a monk. He led Church reform. For him, the Byzantine elegance of Cluny was a distraction from God. The other person was Suger, born poor and given to a monastery by his father. Monastic life proved to be Suger's path into the world, not away from it.

Bernard became the leading theologian of his time; Suger became the politically powerful Abbot of the St-Denis monastery. Suger made monastic reforms that satisfied Bernard, but the two were far apart in spirit. Artistic and architectural elegance did not distract Suger from God; it led him to God.

He was quite explicit about that. He said we could come to understand absolute beauty, which is God, only through the effect of beautiful things on our senses. He said, "The dull mind rises to truth through that which is material."

So, late in life, he set in motion the construction of a new kind of church building. It had ribbed vaults, pointed arches, and exterior buttresses -- spires, stained glass windows, and interior light. In 1137 his people began reshaping the church of St-Denis into the first full-blown Gothic structure.

It was less flagrantly ornate than Cluny -- less massive in appearance. In the very lightness of its being, it summoned up some vestige of the clean austerity that Bernard wanted. Yet it was, in fact, the grandest single step forward that architecture has ever seen. And for Suger, it was also a theological statement.

Geometrical harmony, he said, is the source of all beauty because it exemplifies the laws by which divine reason made the universe. Suger probably never designed anything himself, but he was a fine arbiter of design. And you need only walk through one of those remarkable cathedrals to believe, with him, that material beauty is a glimpse of God’s face.

But back to the idea of structure: Suppose you're a contractor asked to design and erect a quarter-mile-long structure, as tall as a typical downtown office building. Then you're told that it’s to be made of stone alone, no steel beams, no concrete -- that the only available power will be at ground level, limited to horses, oxen, and maybe a three-horsepower water-driven saw. One more thing, you've got to do the job with no stress calculations and without working drawings.

It sounds impossible. Yet that's what the medieval cathedral builders did. The master mason was architect and builder rolled into one. He often directed a work force numbering into hundreds. But he also worked among his people. He cut stone and installed plumbing. That puzzles us, wed as we are to the notion that academic and manual knowledge don't mix. Big mistake!

The subtle grace of the Gothic cathedral touches us powerfully on so many levels. And its grace is vested in analytical design: barrel vaults, flying buttresses, Gothic arches, and spiral stone staircases -- all had to be born of mathematics. Rouen Cathedral

But mathematics means handling numerical quantities symbolically -- something medieval masons did not study. In fact, some couldn't even read. As we comb the rich medieval record, we find no mathematical basis for these glorious buildings; nor do we find any architectural drawings -- only the crudest sketches.

Yet the medieval cathedral is a symphony of geometry and proportion -- from labyrinths in mosaic floor tiles to the crisscrossing ribs that hold the ceiling. It just doesn't make sense.

Then we realize: The building is the geometry text. The master mason, with his fingers touching stone, used stone to express geometry. If mathematics is the symbolic expression of magnitude, that's what the cathedral itself is. The balance of mass and space goes by square roots of 2 and 3, and the so-called Golden Section.

Medieval iconography shows one mathematical instrument in the hands of the mason -- a pair of dividers. When medieval artists show us God, He often appears as the Master Craftsman holding dividers. With that and a carpenter's square alone, one can prove the Pythagorean Theorem; one can create any of those seemingly sacred proportions. The cathedrals were not designed by mathematics; they are mathematics -- mathematics that flowed straight from the mind's eye to the hands of masons who built them.
So the Gothic Cathedral turns our word structure on its ear. At first, we want to distinguish between structure as a material thing or a mental thing. But, as we walk the long nave of one of those stunning buildings, that distinction blurs and it finally disappears entirely.